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Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$ (in affine coordinates) defined by $$y^2=x^3+x.$$ Clearly the discriminant of $E$ is $-2^6$. And it is known that $$ \mid\{(x,y):0<x,y<p,\ y^2\equiv x^3+x\pmod p\}\mid=p-1, $$ where $\mid S\mid$ is the cardinality of a set $S$. However, it seems not obvious to get the explicic value of $$\mid\{(x,y): 0<x<p/2,0<y<p, y^2\equiv x^3+x \pmod p\}\mid.$$ Are there any references on this topic? Your comments are welcome. Thank you.

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    $\begingroup$ This looks like a homework question. If you want a hint, consider the action of the automorphism $(x,y)\mapsto(-x,\sqrt{-1}y)$ (which is not defined over $\mathbb{F}_p$). $\endgroup$
    – Ben Smith
    Commented Nov 23, 2020 at 13:45

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